Connectivity and homology of finite subset spaces of cardinality at most four

The aim of this paper is to study topological features of Sub 4 X ; the finite subset space of cardinality at most four associated to a topological space X. We determine its connectivity in terms of the connectivity of X, hence answering a conjecture of Christopher Tuffley in that case. We compute the Euler characteristic of Sub 4 X for any topological space X. We provide complete homology computations and determine the rational homotopy type of Sub 4 S d for any d-dimensional sphere.